Question

Prove that two odd integers whose difference is 32 are coprime.

Short answer

The integers are coprime since GCD of the difference (32) is 1.

Step-by-step solution

- Define Odd Integers

Let's assume the two odd integers are \( a \) and \( b \). By definition, any odd integer can be written in the form \( 2k + 1 \) where \( k \) is an integer. So, let the first odd integer be \( a = 2m + 1 \) and the second be \( b = 2n + 1 \) for some integers \( m \) and \( n \).

- Express the Difference

Given that the difference between the two odd integers is 32, we express this as: \( a - b = 32 \). Substituting the expressions for \( a \) and \( b \), we get:\[(2m + 1) - (2n + 1) = 32\] which simplifies to:\[2m + 1 - 2n - 1 = 32\] \[2m - 2n = 32\] \[2(m - n) = 32\] \[m - n = 16\].

- Evaluate for Coprime Property

Two numbers are coprime if their greatest common divisor (GCD) is 1. We need to show that \( a \) and \( b \) written as general forms \( a = 2m + 1 \) and \( b = 2n + 1 \) with \( m-n = 16 \) are coprime. Substituting \( b = a - 32 \) into the general forms gives us that \( b = a - 32 \) still takes odd integer form. Since neither includes odd factors in any linear arrangement of their values, and the difference is even and non-zero, their respective terms do not inherently overlap in any factor other than \( 1 \).

- Apply Number Theory

If two numbers are not coprime, they share a common divisor greater than 1. Since both integers are odd and differ by an even common multiple of 16, assume a common divisor \( d > 1 \) dividing them:- Then \( d \) must divide their difference 32.- The only odd divisor of 32, since only odd \( d \) would fit both \( a \) and \( b \), by lack of even factor, is 1.- Therefore, the GCD is 1.

- Conclusion

Given the constraints and simplifications, no odd divisor of a positive power of even (32) divides both odd expressions other than 1. This confirms that both integers are coprime.

Key concepts

Odd Integers

Odd integers are numbers that cannot be evenly divided by 2. They take the form of 2k + 1, where k is any integer.
For example, when you substitute k with 0, 1, 2, 3, and so on, you get the sequence of odd integers like 1, 3, 5, 7, etc.
The sequence retains an interval of 2, so each number's gap between each other remains consistent.

• They always have a remainder of 1 when divided by 2.
• The operation of adding or subtracting an even number does not change the oddness of a number.
• For two odd numbers a and b, their difference (a-b) will always be even.

Understanding these characteristics helps when manipulating and comparing odd integers, as in our exercise where we found that two odd integers differed by 32.

Coprime Numbers

Coprime numbers are two or more numbers whose greatest common divisor (GCD) is 1. They do not share any common factors other than 1, even though individually they might not be prime.
Coprimality doesn't require numbers to be sequential or paired in any particular way.

• If x and y are coprime, the only positive integer that divides both is 1.
• This is still the case even if numbers themselves are complex, as long as they share no even factor overlap.

In our given exercise, even though both numbers are odd and differ by 32, we can conclude their coprimacy through structured derivations, like assuming no shared divisor beyond unity exists. This makes checking coprime status through their integer forms reliable.

Greatest Common Divisor

The greatest common divisor (GCD) of two numbers is the largest integer that evenly divides both numbers. It is crucial for understanding the divisible relationship between numbers.
Finding the GCD often involves identifying any common factors between the numbers or using the Euclidean algorithm.

• The GCD of numbers a and b, written as GCD(a, b), forms the baseline condition needed to prove coprimacy.
• If the GCD is 1, the numbers are coprime.

Our exercise showed that assuming any divisor other than 1 for the odd integers that differed by 32 didn't hold, reinforcing the property that their only shared divisor was indeed 1.